Missing data is frequently encountered in many areas of statistics.
Imputation and propensity score weighting are two popular methods for handling
missing data. These methods employ some model assumptions, either the outcome
regression or the response propensity model. However, correct specification of
the statistical model can be challenging in the presence of missing data.
Doubly robust estimation is attractive as the consistency of the estimator is
guaranteed when either the outcome regression model or the propensity score
model is correctly specified. In this paper, we first employ information
projection to develop an efficient and doubly robust estimator under indirect
model calibration constraints. The resulting propensity score estimator can be
equivalently expressed as a doubly robust regression imputation estimator by
imposing the internal bias calibration condition in estimating the regression
parameters. In addition, we generalize the information projection to allow for
outlier-robust estimation. Thus, we achieve triply robust estimation by adding
the outlier robustness condition to the double robustness condition. Some
asymptotic properties are presented. The simulation study confirms that the
proposed method allows robust inference against not only the violation of
various model assumptions, but also outliers